Both planets are in motion, so the distance between them varies considerably. There are two extremes:
Firstly, Earth is furthest from the Sun (aphelion, 152,097,701 km) and Neptune is in opposition (directly opposite the Sun in the sky so that the Earth is exactly between the Sun and Neptune) and nearest to the Sun (perihelion, 4,452,940,833 km). The separation is then 4,452,940,833 km - 152,097,701 km or 4,300,843,132 km. Since 1 light year is 9,460,730,472,580.8 km, Neptune will be 0.0004546 of a light year away. This is the closest possible.
Secondly, Earth is at aphelion and Neptune is at aphelion but in conjunction - diametrically opposite the Earth on the other side of the Sun. In this case the separation will be 4,553,946,490 km + 152,097,701 km or 4,706,044,191 km. This is 0.0004974 of a light year, the furthest possible.
These simplified calculations do not take account of the 1.768° orbital inclination of Neptune.
https://en.wikipedia.org/wiki/Neptune
https://en.wikipedia.org/wiki/Earth%27s_orbit
https://en.wikipedia.org/wiki/Light_year
